How can particles be emitted from particles?
The trick is, you use the particle instance modifier on one particle system, and the particle system is recognized as one object, so it can then be emitted from itself.


Create a baseObject (that will be emitting the particles that emit the particles).
Create a ps1Object (it will be the object that is emitted in the first particle system)
Give the baseObject a particle system.
Select the ps1Object, and give it a particle instance modifier. In the object tab, select baseObject. Make sure to reset the location of the ps1Object, so it looks like it's emitting from the baseObject.
Give the ps1Object a particle system and check Use Modifier Stack in the Emission tab.